Number of Singular Fibers of Pencil of Curves in Projective 3-Space

Question

Let $S \subset \mathbb{P}_{\mathbb{C}}^3$ be a general surface of degree $d$. Consider the pencil of curves on $S$ arising by intersecting $S$ with a general pencil of $2$-planes. How many of the curves in the resulting pencil are singular?

Answer 1

This is $d(d-1)^2$, and comes from a more general formula proven in 3264 and all that by Harris and Eisenbud, chapter 7.

In general, given a general surface of degree $d$ in $\mathbb P^3$, and a generic pencil of surface of degree $e$, you will obtain $d(3e^2 + 2(d-4) + d^2 - 4d + 6)$ singular curves.